Question

Find the value of \[\sin\dfrac{\pi }{{14}}\sin\dfrac{{3\pi }}{{14}}\sin\dfrac{{5\pi }}{{14}}\sin\dfrac{{7\pi }}{{14}}\sin\dfrac{{9\pi }}{{14}}\sin\dfrac{{11\pi }}{{14}}\sin\dfrac{{13\pi }}{{14}}\].
A. \[\dfrac{1}{8}\]
B. \[\dfrac{1}{{16}}\]
C. \[\dfrac{1}{{32}}\]
D. \[\dfrac{1}{{64}}\]

Answer 1

Hint:First, rewrite the terms of the given trigonometric expression by using the various trigonometric identities of the complementary and supplementary angles. Then by using the double angle formula \[2\sin\theta \cos\theta = \sin2\theta \] solve the equation and get the required answer.

Formula used:
\[\sin\left( {\dfrac{\pi }{2} - \theta } \right) = \cos\theta \]
\[\sin\left( {\pi - \theta } \right) = \sin\theta \]
\[\sin\left( {\pi + \theta } \right) = - \sin\theta \]
\[\cos\left( {\dfrac{\pi }{2} - \theta } \right) = \sin\theta \]
\[\cos\left( {\pi - \theta } \right) = - \cos\theta \]
\[2\sin\theta \cos\theta = \sin2\theta \]

Complete step by step solution:
The given trigonometric expression is \[\sin\dfrac{\pi }{{14}}\sin\dfrac{{3\pi }}{{14}}\sin\dfrac{{5\pi }}{{14}}\sin\dfrac{{7\pi }}{{14}}\sin\dfrac{{9\pi }}{{14}}\sin\dfrac{{11\pi }}{{14}}\sin\dfrac{{13\pi }}{{14}}\].
Let \[V\] be the value of the expression.
\[V = \sin\dfrac{\pi }{{14}}\sin\dfrac{{3\pi }}{{14}}\sin\dfrac{{5\pi }}{{14}}\sin\dfrac{{7\pi }}{{14}}\sin\dfrac{{9\pi }}{{14}}\sin\dfrac{{11\pi }}{{14}}\sin\dfrac{{13\pi }}{{14}}\]
Let’s simplify the above equation.
\[V = \sin\dfrac{\pi }{{14}}\sin\dfrac{{3\pi }}{{14}}\sin\dfrac{{5\pi }}{{14}}\sin\dfrac{\pi }{2}\sin\left( {\pi - \dfrac{{5\pi }}{{14}}} \right)\sin\left( {\pi - \dfrac{{3\pi }}{{14}}} \right)\sin\left( {\pi - \dfrac{\pi }{{14}}} \right)\]
\[ \Rightarrow \]\[V = \sin\dfrac{\pi }{{14}}\sin\dfrac{{3\pi }}{{14}}\sin\dfrac{{5\pi }}{{14}}\left( 1 \right)\sin\left( {\dfrac{{5\pi }}{{14}}} \right)\sin\left( {\dfrac{{3\pi }}{{14}}} \right)\sin\left( {\dfrac{\pi }{{14}}} \right)\] [since \[\sin\dfrac{\pi }{2} = 1\] and \[\sin\left( {\pi - \theta } \right) = \sin\theta \]]
\[ \Rightarrow \]\[V = \sin\dfrac{\pi }{{14}}\sin\dfrac{{3\pi }}{{14}}\sin\dfrac{{5\pi }}{{14}}\sin\dfrac{{5\pi }}{{14}}\sin\dfrac{{3\pi }}{{14}}\sin\dfrac{\pi }{{14}}\]
\[ \Rightarrow \]\[V = {\left[ {\sin\dfrac{\pi }{{14}}\sin\dfrac{{3\pi }}{{14}}\sin\dfrac{{5\pi }}{{14}}} \right]^2}\]
Further simplify the above equation.
\[V = {\left[ {\sin\left( {\dfrac{\pi }{2} - \dfrac{{3\pi }}{7}} \right)\sin\left( {\dfrac{\pi }{2} - \dfrac{{2\pi }}{7}} \right)\sin\left( {\dfrac{\pi }{2} - \dfrac{\pi }{7}} \right)} \right]^2}\]
\[ \Rightarrow \]\[V = {\left[ {\cos\left( {\dfrac{{3\pi }}{7}} \right)\cos\left( {\dfrac{{2\pi }}{7}} \right)\cos\left( {\dfrac{\pi }{7}} \right)} \right]^2}\] [since \[\sin\left( {\dfrac{\pi }{2} - \theta } \right) = \cos\theta \]]
\[ \Rightarrow \]\[V = {\left[ {\cos\left( {\pi - \dfrac{{4\pi }}{7}} \right)\cos\left( {\dfrac{{2\pi }}{7}} \right)\cos\left( {\dfrac{\pi }{7}} \right)} \right]^2}\]
\[ \Rightarrow \]\[V = {\left[ { - \cos\left( {\dfrac{{4\pi }}{7}} \right)\cos\left( {\dfrac{{2\pi }}{7}} \right)\cos\left( {\dfrac{\pi }{7}} \right)} \right]^2}\] [since \[\cos\left( {\pi - \theta } \right) = - \cos\theta \]]
Let's consider \[\dfrac{\pi }{7} = \theta \].
Then,
\[V = {\left[ { - \cos\left( x \right)\cos\left( {2x} \right)\cos\left( {4x} \right)} \right]^2}\]
Multiply and divide the right-hand side of the above expression by \[2\sin x\].
\[V = {\left[ { - \dfrac{1}{{2\sin x}}\left( {2\sin x cosx} \right)\cos2x \cos4x} \right]^2}\]
Simplify the equation.
\[V = {\left[ { - \dfrac{1}{{2\sin x}}\sin2x\cos2x\cos4x} \right]^2}\] [since \[2\sin\theta \cos\theta = \sin2\theta \]]
Multiply and divide the right-hand side of the above expression by 2.
\[V = {\left[ { - \dfrac{1}{{4\sin x}}\left( {2\sin2x\cos2x} \right) \cos4x} \right]^2}\]
Simplify the above equation.
\[V = {\left[ { - \dfrac{1}{{4\sin x}}\sin4x\cos4x} \right]^2}\] [ Since \[2\sin\theta \cos\theta = \sin2\theta \]]
Multiply and divide the right-hand side of the above expression by 2.
\[V = {\left[ { - \dfrac{1}{{8\sin x}}\left( {2\sin4x\cos4x} \right)} \right]^2}\]
\[ \Rightarrow \]\[V = {\left[ { - \dfrac{1}{{8\sin x}}\left( {\sin8x} \right)} \right]^2}\]
Now simplify the above equation.
Resubstitute the value of \[x\].
\[V = {\left[ { - \dfrac{1}{{8\sin\dfrac{\pi }{7}}}\left( {\sin\dfrac{{8\pi }}{7}} \right)} \right]^2}\]
\[ \Rightarrow \]\[V = {\left[ { - \dfrac{1}{8}\left( {\dfrac{1}{{\sin\dfrac{\pi }{7}}}\sin\left( {\pi + \dfrac{\pi }{7}} \right)} \right)} \right]^2}\]
Now apply the identity \[\sin\left( {\pi + \theta } \right) = - \sin\theta \]
\[V = {\left[ { - \dfrac{1}{8}\left( {\dfrac{1}{{\sin\dfrac{\pi }{7}}}\left( { - \sin\dfrac{\pi }{7}} \right)} \right)} \right]^2}\]
Simplify the above equation.
\[V = {\left[ { - \dfrac{1}{8}\left( { - 1} \right)} \right]^2}\]
\[ \Rightarrow \]\[V = {\left[ {\dfrac{1}{8}} \right]^2}\]
\[ \Rightarrow \]\[V = \dfrac{1}{{64}}\]
Hence the correct option is D.

Note: The cofunction identities describe the relationship between trigonometric functions sine, cosine, tangent, cotangent, secant and cosecant. The value of a trigonometric function of an angle equals the value of the conjunction of the complement.